An infinite-dimensional phenomenon in finite-dimensional metric topology

We show that there are homotopy equivalences h : N -> M between closed manifolds which are induced by cell-like maps p : N -> X and q : M -> X but which are not homotopic to homeomorphisms. The phenomenon is based on the construction of cell-like maps that kill certain IL-classes. The image space in these constructions is necessarily infinite-dimensional. In dimension > 5 we classify all such homotopy equivalences. As an application, we show that such homotopy equivalences are realized by deformations of Riemannian manifolds in Gromov-Hausdorff space preserving a contractibility function.